Average Error: 32.4 → 22.6
Time: 19.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{2 \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\frac{2 \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}
double code(double t, double l, double k) {
	return (2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
}
↓
double code(double t, double l, double k) {
	return (((2.0 * (pow((cbrt(1.0) / pow((cbrt(t) * cbrt(t)), (3.0 / 2.0))), 1.0) * (pow((cbrt(1.0) / pow((cbrt(t) * cbrt(t)), (3.0 / 2.0))), 1.0) * (pow((cbrt(1.0) / pow(cbrt(t), 3.0)), 1.0) * (l / sin(k)))))) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0))));
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

    \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

  2. Simplified32.4

    \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity32.4

    \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

  5. Applied times-frac31.5

    \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

  6. Applied associate-*r*28.7

    \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

  7. Simplified27.6

    \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  8. Taylor expanded around inf 27.4

    \[\leadsto \frac{\color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt27.7

    \[\leadsto \frac{2 \cdot \left({\left(\frac{1}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  11. Applied unpow-prod-down27.7

    \[\leadsto \frac{2 \cdot \left({\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  12. Applied add-cube-cbrt27.7

    \[\leadsto \frac{2 \cdot \left({\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  13. Applied times-frac27.5

    \[\leadsto \frac{2 \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}}^{1} \cdot \frac{\ell}{\sin k}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  14. Applied unpow-prod-down27.5

    \[\leadsto \frac{2 \cdot \left(\color{blue}{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot {\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1}\right)} \cdot \frac{\ell}{\sin k}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  15. Applied associate-*l*24.0

    \[\leadsto \frac{2 \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
  16. Using strategy rm

  17. Applied sqr-pow24.0

    \[\leadsto \frac{2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  18. Applied times-frac23.9

    \[\leadsto \frac{2 \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  19. Applied unpow-prod-down23.9

    \[\leadsto \frac{2 \cdot \left(\color{blue}{\left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot {\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1}\right)} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  20. Applied associate-*l*22.6

    \[\leadsto \frac{2 \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

  21. Final simplification22.6

    \[\leadsto \frac{2 \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))